Finite-time thermodynamics asks: what is the minimum work required to transform a system from one equilibrium state to another in a given time? The question seems well-defined. The answer depends on what you mean by “minimum.”
Speed limits on the control protocol are essential to a consistent formulation (arXiv:2603.11995). Without speed limits, the optimal protocol can change infinitely fast — instantaneous jumps in temperature, potential, or pressure — and the “minimum work” reduces to an artifact of allowing unphysical control. The speed limit is not a practical constraint added for engineering convenience. It is a theoretical requirement for the problem to be well-posed.
With speed limits, two distinct optimal strategies emerge: swift engineered equilibration (driving the system through a sequence of near-equilibrium states as quickly as the speed limit allows) and minimum work transitions (minimizing dissipation given the speed constraint). These are different optimizations with different solutions. Without speed limits, they conflate — both converge to the same instantaneous protocol, hiding the distinction.
Only generalized Schrodinger bridges maintain consistent physical interpretation when speed limits are removed. The Schrodinger bridge — originally a quantum mechanical concept applied to stochastic processes — provides a natural framework because it minimizes a relative entropy (a measure of how far the actual path deviates from the unconstrained path) rather than work directly. The bridge formulation remains well-defined even at infinite control speed because it optimizes a path-level quantity rather than a pointwise one.
The minimum work question is only meaningful when you specify how fast you're allowed to act.